In the present paper we study quasi-Monte Carlo rules for approximating integrals over the $d$-dimensional unit cube for functions from weighted Sobolev spaces of regularity one. While the properties of these rules are well understood for anchored Sobolev spaces, this is not the case for the ANOVA spaces, which are another very important type of reference spaces for quasi-Monte Carlo rules.
Using a direct approach we provide a formula for the worst case error of quasi-Monte Carlo rules for functions from weighted ANOVA spaces. As a consequence we bound the worst case error from above in terms of weighted discrepancy of the employed integration nodes. On the other hand we also obtain a general lower bound in terms of the number $n$ of used integration nodes.
For the one-dimensional case our results lead to the optimal integration rule and also in the two-dimensional case we provide rules yielding optimal convergence rates.
Equivalence of anchored and ANOVA spaces via interpolation
Institut für Analysis, Johannes Kepler Universität Linz（林茨约翰·开普勒大学分析研究所）；Institut für Mathematik, Friedrich Schiller Universität Jena（耶拿弗里德里希·席勒大学数学研究所）
In this paper we propose an approximation method for high-dimensional $1$-periodic functions based on the multivariate ANOVA decomposition. We provide an analysis on the classical ANOVA decomposition on the torus and prove some important properties such as the inheritance of smoothness for Sobolev type spaces and the weighted Wiener algebra. We exploit special kinds of sparsity in the ANOVA decomposition with the aim to approximate a function in a scattered data or black-box approximation scenario. This method allows us to simultaneously achieve an importance ranking on dimensions and dimensio…
We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space on $\mathbb{R}^d$, for $d \geq 1$. Both spaces are Hilbert spaces involving weight functions, which determine the behaviour as different variables tend to $\pm \infty$, and weight parameters, which represent the influence of different subsets of variables. The unanchored ANOVA space on $\mathbb{R}^d$ was initially introduced by Nichols & Kuo in 2014 to analyse the error of quasi-Monte Carlo (QMC) approximations for integrals o…
A Christoffel function weighted least squares algorithm for collocation approximations
University of Utah（犹他大学）；Sandia National Laboratories（桑迪亚国家实验室）；Institute of Computational Mathematics and the Chinese Academy of Sciences（中国科学院计算数学与科学工程计算研究所）
The Stochastic Weighted Particle Method (SWPM) is a Monte Carlo technique developed by Rjasanow and Wagner that generalizes Bird's Direct Simulation Monte Carlo (DSMC) method for solving the Boltzmann equation. To reduce computational cost due to the gradual increase in the number of stochastic particles in the SWPM, Rjasanow and Wagner proposed several particle reduction schemes designed to preserve specified moments of the velocity distribution. Here, we introduce an improved particle reduction scheme that preserves all moments of the velocity distribution up to the second order, as well as …
In this note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions defined over the $s$-dimensional unit cube by using rank-1 lattice point sets only for the first $d\, (<s)$ coordinates and random points for the remaining $s-d$ coordinates. We prove that, by exploiting a decay of the weights of function spaces, almost the optimal order of the mean squared worst-case error is achieved by such a concatenated quadrature rule as l…
We study the efficient construction of good polynomial lattice rules, which are special instances of quasi-Monte Carlo (QMC) methods. The integration rules obtained are of particular interest for the approximation of multivariate integrals in weighted Walsh spaces. In particular, we study a construction algorithm which assembles the components of the generating vector, which is in this case a vector of polynomials over a finite field, of the polynomial lattice rule in a component-wise fashion. We show that the constructed QMC rules achieve the almost optimal error convergence order in the func…
Small-noise analysis and symmetrization of implicit Monte Carlo samplers
University of Arizona（亚利桑那大学）；University of California at Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary cube. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution functi…
